The Cauchy Problem of Null Form Wave Equation on T^d with Random Initial Data

抄録

In this paper, we consider the local well-posedness of the wave equation with null form nonlinearities under the periodic boundary condition. First, we prove the local existence based on some L^p-based Sobolev like Space $\hat{H}$^s,p, where s > 1/2 + d/p′ for d ≥ 2, s ≥ 1/2, for d = 1. Second, we prove the local existence for randomized initial data in H^s with s > 1/2 and d ≥ 2.